5

IIIResolve the not-homogeneous differential equation with the parameter variation method. y" _ 4y = 8e2x b_ y" +y sec2(x)...

Question

IIIResolve the not-homogeneous differential equation with the parameter variation method. y" _ 4y = 8e2x b_ y" +y sec2(x)

III Resolve the not-homogeneous differential equation with the parameter variation method. y" _ 4y = 8e2x b_ y" +y sec2(x)



Answers

Solve the differential equation using (a) undetermined coefficients and (b) variation of parameters.

$ y'' - 2y' - 3y = x + 2 $

Hello and welcome to problem three of Chapter three. Section 7 you were asked to use the method of variation of parameters to find a particular solution to the given differential equation. Alright so a different equation in this case is Y double prime plus two? Y. Prime plus Y equals three E. To the minus T. And start this off we're going to find the characters to a solution. And that was his drive from the characteristic equation which is R squared plus two are plus one equals zero here we'll have our +11. He squared equals zero. So R. Is equal to negative one will have double it there. So that means why one equals eat the negative T. And white too. Because T. E. It's the negativity. Okay from these uh we'll drive this capital wipes term which is just what is a what we need to add on to the end because we are not left with the equal zero. We're left with an equals E. To the minus T. Okay. And to find this, there's a simple uh equation for this. Well not simple. It's rather long tedious. It's negative. Why one times integral of Y two. GFT were GFT is just this turn here. Over the wrong skin of wise. You can get to that later. D. T. Plus the integral of Oops, Y. two times the integral of why one GFT through the wrong skin dems duty. Okay now what is the run down in this case? Well let's write Ron skin and blue. And that's the two x 2 determinants of its solutions of why 1? Why two by one prime. Why to prime? And in this case This is equal to one times y to prime. Well that's Um either the -T times the derivative of Y two. Get to that in a second. Um Let's go DDT TE -2. Where to subtract this from? Um why to which is to either -1 attempts the derivative of why one prime does negative E to the negative T. So I'm just gonna add like this E to the minus to you. Okay from here we have to find the derivative of T to the minus T. Um There's a few ways to do this. Um But this is really just product rule. So although um Derivative of this is T. 10 to derivative of E- T. is negative. Yeah. The ***. Yeah, negative E to the negative too. And we're going to add this to even the negativity Times The Derivative Tier, which is one. Okay, and if we simplify this some more, we got blue again. Well E to the -2 times either minus two minus T. Either minus two. Yeah. Plus T. E to the -2. T. Keep simplifying. We'll get E to the -2 T minus T. E to the -2 T. Of course this will cancel that with this there. So we'll just be left with E. to the -2 to. Okay. And we can plug everything back into this big Y. Of T. Expression here. Uh right there. Okay so big Y. Of T. Equals negative Y. One. That's a negative view of the -2 comes into rule of Y. Two times. Gmt so Y. Two T. E. To the minus T. G. F. T. S. Three E. To the minus two Over the run skin is E. to the -2 T. Then we're gonna add this to um why too? Uh T. E. To the minus T. Mhm. Who is the integral Of? Why 1? Either? The- T. Times G. O. T. Which is three E. To the minus two Divided by rounds skin which is e. to the -2 T. D. T. Okay from here we'll just simplify and solve for Y. Of T. Uh Looking at this you can see that we'll have an E. To the minus two T. On top and either the mine institute on the bottom. So that will cancel. Cantel cancel, disease will cancel here. Whoops. We'll have Y. Of T. Equal negative et minus T. The integral of T. D. T. And allowed this to T. E. To the minus T. Times the integral of three D. T. Well I'm just going to Um quickly make this 32. Okay and this integral of tea is just one half T squared. So I'll change that one half T. Squared. Okay uh As you can see these two terms can't be combined so we'll have to leave it as is but we want to simplify one more time. Yeah. Oh yes, they can be combined, apologize for that. I will have um three t squared either the minus t minus t squared E to the minus two over to. Yeah. And that will simplify one last time to now. Five t squared E to the minus T over to. So final solution. This entire difference equation, how do we know Y of T is Y equals C1 times either minus two plus C two T to the -2 plus five T squared E to the minus T over to. And that concludes the problem.

Today we have this problem which we're going to solve using the method of variation of parameters. This is a second our linear No, no Imogen 70 we will denote the right now. Zero rare inside by F X And as usual, the first thing we need to do is to solve thie How much years in question, which is simply the same in creation, with the right hand side being simply zero and something. This is very simple. We simply write down the character see equation that isthe r squared plus three are prostituting Ciro. And once you solve this, we will have and two solutions I want to say the one and art to its new two, which means that our two annual independent solutions for damages parts are why one is able to Internet e X, and why two is Turner to X. So we have This means we have these general solution for the generous part which would threaten us. Why want see one white one must see to why two So the only thing that's missing now is thie is way. We have to find a particular solution ritual call by wise api and they and the method of fishing parameters tells us that we need to look for this guy and turns off a round in the condition ofthe into solution we've just obtained. So we're looking for two functions experience such then this. Ah, this baby thing is a solution off this equation so we could go through all the algebra off by taking on the drilling through spread. Fortunately for us, there's a very well known formula for each of this expects. So we we can simply use that Oh ah talked in our solution. So they form a line I will, which I will write down here. So a off X is simply equal to negative and tiger to off Why two times f f x, the ex and derided by what we call the bone scan off our two functions. They reverted found so and so Let's continue this actually, so you know, in order for this, we need to find the Royce can. So, as you might recall from the previous chapter y one, if you have two functions, they run a scan is equal to its two by two insurance. Why want prime word to prime Andi that's convenient. This on the next page. So this is equal to remember why one y two were into a negative axe on DH to marry to X. So they're doing. There's our native E to the native Axe and daters Later, two times easy to XO computer this to writer determined. You'll see that this is equal to native each Max. So what we have to do now is simply Valerie dissented during. So now we know all the ingredients y two ever x, which is this? And Delhi. So it's student here. So ate off Max. Cool. Two negative and center itself. So I have y two times f here, so there will be too native to x times Signe Well, you need to wax the Ryan. We're in the wrong skin, which is just this guy here. So it's again negative. The 23 acts Yes, here. So after the nature science cancel and some of the expansion terms can so we see that this is equal to and danger into off e two ex Ein teo angsty axe. Now this. Now, this is a simple cal calculus to integral and their several ways to do this But you can probably. Or to see that if we use the u substitution with around with this. But this is a different Grable tear. So if assassinated the expression lexical to tea, then receive it. The tea is equal to the axe. Bruce, Charlie yaks the X so that this is a simply equal to sign. Sign it, t two. This is because our DT contains this eatery Exterminate! Sign of X. Simple becomes sanity. So now this is very simple. It's simply Nate. Of course I am too right away. So you don't have to put plus C here because since we're so if you put plus some see here than in the end, we write down the answer. We have to multiply this by why one So. So do you concede that this term's constant times? Wyman term is already, you know, taking care of and the have a homogeneous solution. So that's why we don't have to deal with plus c constant here. So that's just something to remember. And so let's move on to the calculation of BX. So it's just that it's sir a new bitch here. Yeah, so b X, it's very spit it taking away the same chocolate. So this is able to refuel? No. Well, and I guess I haven't written out from was its recently consumed you simply If this native sign and this will be why want so e x would be and endure it off? Why? One times f of acts times uh, the dumping. So why one was Inter native Axe, This is Signe off ee to axe DX, derided by nature's being there three x so after cancellation, this becomes negative. Two to x Sign of X sanity to ex Deeks. Now this already. Look, this is already looking very similar to you. Just that way won't simply that Substitute TV because eat axe so that we have did too sick or too e t x t x and so it will have negative. We'll have negative. So you see that we have each other to X here. So one So this will become t fire sign team T t. Because one one'd rex comes from DT part and one eater X from t part. So they so we'll have our three Turks. So now for here this again is a routine exercise for the integration private by parts method. So if we didn't have this extra t here than we could directly involved at this. So our goal is to get rid of this team by taking its through into which is one. So suggests that we think we do the following intuition re parts. Um, you is in court to Teo e sitting cool too. Sign off, too. Do too, right? Actually, it's put in their sign here. Jasper Hammond, later scientist. So then this would mean the U. S. Simply Teo. They were due to have tea is just once. So take a look at this one here and so d From this, we can see the V is simply co sign off team because director of co sign is nature sign. So no, no. So now we'll have This is simply you times fee, which is two times coursing. Tio the time score sign to you. Minus three times you sort This would be integration course. Sign off too. Okay, team. And that this is very simple. It's just entering. Dissented, and it is just a sign of so some rest that here now have tea co sign two minor sign two and remember to see their acts. So we had to down invert over back to our x variable. So, you know, I have a sign off the ax or dismember. This is R E X so we can write down our oh social y p now. So this is number. So this is X times y one plus beer sounds right here. So why one and way too, Zaheer and a access their co sign t y gets different here too. That right? Everything in terms ofthe x variable. So this is my X. So why, Why he would be Why? So it's just right down here. X y one less yaks. Why two on X is Native Curse sign being axe tires. He terminated axe and be Xs here and multiply that by right eternity to excellence Prentice's Yeah, thanks. Cor sign. Really secure co sign? Yeah, thanks Minus sign off. The X times are y two witches Need two tubes now. Interested? There's usually calculation. So it's It's recommended that when of whatever you've got here, you should open the bracket sent and look for cancer. And indeed, in our case, we have one because we have three X here and Internet to exterior ritual becoming through Native X Times, Carson E X, which will exactly cast with. So I'm going to use a different card to emphasize the cancellation. So this term will cancel with this and this which will leave us with only one term. That is negative E too. Two X tires sign off ee to ACS. So this is my particular station wide piece. So if I write down two final answer, you will be simply Why single too? I see. That's why particular c one e during native X plus c too inter tuner to x minus. Um, well, class white people in this was minus dysfunction Certs minus E to native to x times. Sign off the X and this will be the final Astor for crawl.

Hello. So today we're gonna work on this problem. And so what it is is the second derivative of Y minus two Y, uh, minus two times the first derivative of y plus why is equal to e to the power of two X? So, as you can see, this equation has two parts to it. There's the left side. And then there's the right side. Now that the right side has just a zero like only a zero, this would be a homogeneous differential equation. But because there is, uh, something on the right side, it is not homogeneous in the solution. Will have two parts to it. So the solution since it has two parts, it'll look like this. Why of X is equal to why sea of X, which is the general part of the solution. Plus why P of X, which is a particular part of the solution. So the general part of the solution goes along with the left side of the equation the blue side and ah, particular side. The solution goes with the yellow side. Eso What we're gonna do is first, we're going to solve for the left side part of the solution to do this, we're going to get We're going to derive a complimentary equation to help us find a solution. And so, the way we're going to do this, we're gonna use the variable r, and then we're going to copy the coefficients of the blue side of the problem. So if this first coefficient is one right, then it's going to be one times r squared. And so, as you see, we're gonna use the variable art, and it's gonna go down by degrees. Okay, so next we have minus two. That's a coefficient, right? So minus two. And then this is only our And then here we have a coefficient of one. So will be plus one positive one. We're gonna set this equal to zero, and then we're gonna solve for R and find the root, which in this case, the way we're gonna do this, um, are is going to be equal to one. So there's only one root for this general side of the equation, the blue side. And when there's only one route Ah, there's a fear. Um, that states that when there's only one route the general part of the solution well, look like this is gonna be see one times e to the power of our X. That's an art. Plus the two times x times e to the power of our So that's an art. Now, if there were two routes, uh, it would be R one and r two been this case r equals one. So actually, one times X, it's just x. And so one time sex is just sex. So this will actually be the blue part. Ah, of the solution. So that is this part, right? Nice. So now we can move on to the yellow part of the solution. The particular part of the solution. So we're gonna do this is we're gonna say y p of X is equal to see a times E to the power of K X. So if we go up here, we can see we can copy, uh, C equals one, right? And then K equals two, right, because she is just one and then or C is just one listen more like this, and then K equals two. So why P of X is actually equal to just a times E to the power of two x. Okay, so we need to actually solve for a and to do that, we're gonna find the derivatives of Wipe. So the first derivative of I P is actually going to be too times a times E to the power of two X and the derivative of the second derivative of wipe is actually gonna be four times a times e to the power of two X. So here we have three different things, right. We have YPF X. We have the first derivative of Y p of X, and then we have the second hoops. We're gonna have the second derivative of Y P of X. So there are three things that we got there and what we're going to do with this information is actually going to copy the original format of a problem. And we're gonna substitute in why and the first derivative of y and the second derivative of y. So what that's gonna look like we're just gonna have the second derivative of why which is for a times E to the power of two X and then we're gonna follow this right, we're gonna follow it. So it's negative two times uh, the first derivative which is to a times e to the power of two X And then we're gonna add just why, Right? Uh, we're gonna add y, which is a times E to the power of two X and we're gonna set that all equal to e to the power of two X and we're gonna solve for a And as you can tell, it's for a time. It's a part of two x minus and then this is going to turn into four a times e to the power of two X So it's gonna be four minus four. Ah, so actually these to cancel out and we're left with only this So a is equal to one and we're gonna substitute that in here. So why P of X, right? Sorry, Y p of X is actually equal to since a is one is just equal to e to the power of two X. So this is actually the second part of the equation. And to get the entire solution, wait to get the entire solution. All of this we have to add this and this together. And so the solution will be why of X is equal Thio. See one times e to the X. We'll see two times x times e to the X plus e to the power of two X and that right there will be your solution.

Hello. So today we're gonna work on this problem. And so what it is is the second derivative of Y minus two Y, uh, minus two times the first derivative of y plus why is equal to e to the power of two X? So, as you can see, this equation has two parts to it. There's the left side. And then there's the right side. Now that the right side has just a zero like only a zero, this would be a homogeneous differential equation. But because there is, uh, something on the right side, it is not homogeneous in the solution. Will have two parts to it. So the solution since it has two parts, it'll look like this. Why of X is equal to why sea of X, which is the general part of the solution. Plus why P of X, which is a particular part of the solution. So the general part of the solution goes along with the left side of the equation the blue side and ah, particular side. The solution goes with the yellow side. Eso What we're gonna do is first, we're going to solve for the left side part of the solution to do this, we're going to get We're going to derive a complimentary equation to help us find a solution. And so, the way we're going to do this, we're gonna use the variable r, and then we're going to copy the coefficients of the blue side of the problem. So if this first coefficient is one right, then it's going to be one times r squared. And so, as you see, we're gonna use the variable art, and it's gonna go down by degrees. Okay, so next we have minus two. That's a coefficient, right? So minus two. And then this is only our And then here we have a coefficient of one. So will be plus one positive one. We're gonna set this equal to zero, and then we're gonna solve for R and find the root, which in this case, the way we're gonna do this, um, are is going to be equal to one. So there's only one root for this general side of the equation, the blue side. And when there's only one route Ah, there's a fear. Um, that states that when there's only one route the general part of the solution well, look like this is gonna be see one times e to the power of our X. That's an art. Plus the two times x times e to the power of our So that's an art. Now, if there were two routes, uh, it would be R one and r two been this case r equals one. So actually, one times X, it's just x. And so one time sex is just sex. So this will actually be the blue part. Ah, of the solution. So that is this part, right? Nice. So now we can move on to the yellow part of the solution. The particular part of the solution. So we're gonna do this is we're gonna say y p of X is equal to see a times E to the power of K X. So if we go up here, we can see we can copy, uh, C equals one, right? And then K equals two, right, because she is just one and then or C is just one listen more like this, and then K equals two. So why P of X is actually equal to just a times E to the power of two x. Okay, so we need to actually solve for a and to do that, we're gonna find the derivatives of Wipe. So the first derivative of I P is actually going to be too times a times E to the power of two X and the derivative of the second derivative of wipe is actually gonna be four times a times e to the power of two X. So here we have three different things, right. We have YPF X. We have the first derivative of Y p of X, and then we have the second hoops. We're gonna have the second derivative of Y P of X. So there are three things that we got there and what we're going to do with this information is actually going to copy the original format of a problem. And we're gonna substitute in why and the first derivative of y and the second derivative of y. So what that's gonna look like we're just gonna have the second derivative of why which is for a times E to the power of two X and then we're gonna follow this right, we're gonna follow it. So it's negative two times uh, the first derivative which is to a times e to the power of two X And then we're gonna add just why, Right? Uh, we're gonna add y, which is a times E to the power of two X and we're gonna set that all equal to e to the power of two X and we're gonna solve for a And as you can tell, it's for a time. It's a part of two x minus and then this is going to turn into four a times e to the power of two X So it's gonna be four minus four. Ah, so actually these to cancel out and we're left with only this So a is equal to one and we're gonna substitute that in here. So why P of X, right? Sorry, Y p of X is actually equal to since a is one is just equal to e to the power of two X. So this is actually the second part of the equation. And to get the entire solution, wait to get the entire solution. All of this we have to add this and this together. And so the solution will be why of X is equal Thio. See one times e to the X. We'll see two times x times e to the X plus e to the power of two X and that right there will be your solution.


Similar Solved Questions

5 answers
A car (S driving on level highway It /s acted upon by Ihe (ollowing forces downwrard gravitational force of 12 kN, an upward contact force due t0 Ihe roud of 12 kN, anolhier conlact force due t0 the rOad of - KN dlrected West; and drag force due to alr resistance ol 5 kN directed East What Is the net force acting On the car?
A car (S driving on level highway It /s acted upon by Ihe (ollowing forces downwrard gravitational force of 12 kN, an upward contact force due t0 Ihe roud of 12 kN, anolhier conlact force due t0 the rOad of - KN dlrected West; and drag force due to alr resistance ol 5 kN directed East What Is the ne...
5 answers
(1 point) Let A = ~6 2] Find S, D,and S such that A = SDS [88] , [F6] 5-1 [88]
(1 point) Let A = ~6 2] Find S, D,and S such that A = SDS [88] , [F6] 5-1 [88]...
5 answers
Use Newton's Iaw for cooling (lecture 2) T(t) = T; + (To TH)e-k to solve the problem: Suppose that a cup of soup is placed in a Freezer where the temperature is ~15"C, and it takes 5 minutes to cool down to 609 C How long would it take the soup to cool from 90'C to 35"C.12.2 min11.9 min15.5 min: 10.7 min11.03 min10.6 min:104 min:
Use Newton's Iaw for cooling (lecture 2) T(t) = T; + (To TH)e-k to solve the problem: Suppose that a cup of soup is placed in a Freezer where the temperature is ~15"C, and it takes 5 minutes to cool down to 609 C How long would it take the soup to cool from 90'C to 35"C. 12.2 min...
5 answers
Complete the proof of Note by showing that $f_{y}=Q(x, y)$.
Complete the proof of Note by showing that $f_{y}=Q(x, y)$....
5 answers
Rank the following substances in order of increasing acidity:
Rank the following substances in order of increasing acidity:...
5 answers
Determine whether each of the following molecules is polar or. nonpolar.a. $mathrm{SiCl}_{4}$b. $mathrm{CF}_{2} mathrm{Cl}_{2}$c. $mathrm{SeF}_{6}$d. $mathrm{IF}_{5}$
Determine whether each of the following molecules is polar or. nonpolar. a. $mathrm{SiCl}_{4}$ b. $mathrm{CF}_{2} mathrm{Cl}_{2}$ c. $mathrm{SeF}_{6}$ d. $mathrm{IF}_{5}$...
5 answers
Examine the geomety ofthe molecule in the Window;ball & stlckIbalsThe molecule has carbony| groupRCHO RCOR" whete and R* are carbon groups. RCOOH ROHA gencral funetional group Fepresentntion Of the molecule iThe compound is a(n) aldohyde
Examine the geomety ofthe molecule in the Window; ball & stlck Ibals The molecule has carbony| group RCHO RCOR" whete and R* are carbon groups. RCOOH ROH A gencral funetional group Fepresentntion Of the molecule i The compound is a(n) aldohyde...
5 answers
Letp = F,9=T=TFor the following propositions, select True or False for its correct evaluationQ4 1 PointP V (9 ^ 7) TrueFalse042 1 Point47) ^ (P 7q) TrueFalseQ433 1 Point0 9) ^-(q er) TrueFalse
Letp = F,9=T=T For the following propositions, select True or False for its correct evaluation Q4 1 Point P V (9 ^ 7) True False 042 1 Point 47) ^ (P 7q) True False Q433 1 Point 0 9) ^-(q er) True False...
5 answers
A 20-V potential difference is applied across a seriescombination of a 10-Ω resistor and a 30-Ω resistor. The current inthe 10-Ω resistor is Question 19 options:a) 2 Ab) 0.5 Ac) 1 Ad) 0.67 AA 20-V potential difference is applied across a seriescombination of a 10-Ω resistor and a 30-Ω resistor. Thepotential difference across the 10-Ω resistor is Question 20 options:a) 10 Vb) 5 Vc) 20 Vd) 15 V
A 20-V potential difference is applied across a series combination of a 10-Ω resistor and a 30-Ω resistor. The current in the 10-Ω resistor is Question 19 options: a) 2 A b) 0.5 A c) 1 A d) 0.67 A A 20-V potential difference is applied across a series combination of a 10-Ω resistor ...
1 answers
Sum of an Infinite Geometric Series, find the sum of the infinite geometric series. $$ -\frac{125}{36}+\frac{25}{6}-5+6-\cdots $$
Sum of an Infinite Geometric Series, find the sum of the infinite geometric series. $$ -\frac{125}{36}+\frac{25}{6}-5+6-\cdots $$...
5 answers
00 00 1 2.If On converges, then converges 0 1+an n [
00 00 1 2.If On converges, then converges 0 1+an n [...
5 answers
Find the intersection of the line and the circle givenbelow.4x+y=-1x^2+y^2=10
Find the intersection of the line and the circle given below. 4x+y=-1 x^2+y^2=10...
5 answers
Provide a detailed, arrow-pushing mechanism for the hydrolysis reaction below. [15 points]OCH3HzOCHzOHHCI cat.OH
Provide a detailed, arrow-pushing mechanism for the hydrolysis reaction below. [15 points] OCH3 HzO CHzOH HCI cat. OH...
5 answers
QUESTIONFor the suspended traffic light what is the ratio of the tensions T1/1T2? (Give answer to at least 3 significant figures)37.0953.0"T
QUESTION For the suspended traffic light what is the ratio of the tensions T1/1T2? (Give answer to at least 3 significant figures) 37.09 53.0" T...
5 answers
Solve the compound inequality and notation: graph the solution~14 <-31+4 < -5(-0,3]U[6,0)(3,6]J0 " "No solution[3,6)Kh
Solve the compound inequality and notation: graph the solution ~14 <-31+4 < -5 (-0,3]U[6,0) (3,6] J0 " " No solution [3,6) Kh...
5 answers
Parallelogram has sides of lengths and and one angle is 459 Find the lengths of the diagonals (Round your answers to two decimal places_ Enter your answers as comma separated list: )
parallelogram has sides of lengths and and one angle is 459 Find the lengths of the diagonals (Round your answers to two decimal places_ Enter your answers as comma separated list: )...

-- 0.029959--